Flow around a Liquid Sphere Filled with a Non-Newtonian Liquid and Placed into a Porous Medium

Abstract

Flow around a Reiner–Rivlin non-Newtonian liquid particle, which is surrounded with a Newtonian liquid shell and placed into a permeable medium, is studied. This formulation of the problem is typical for, e.g., studying the motion of an oil droplet surrounded with an aqueous medium (oil-in-water emulsion) in a porous collector under the action of an external pressure drop. An analogous problem is encountered when lymph penetrates into human or animal tissues. The flows inside of the permeable layer, in the region between the Reiner–Rivlin liquid and a porous medium, and inside of the spherical region are described by the Brinkman, Stokes, and Reiner–Rivlin equations, respectively. The general solution for the stream function in the external porous region is written in terms of the modified Bessel function and Gegenbauer polynomials. For the Reiner–Rivlin liquid sphere, the solution is found by expanding the stream function into a power series in terms of small dimensionless parameter S. The boundary problem is solved by conjugating the boundary conditions for all regions. The drag force applied to the Reiner–Rivlin liquid particle placed into the permeable medium is determined. The effects of permeability parameter α, viscosity ratio λ, and dimensionless parameter S on the drag coefficient are studied. Corresponding dependences are represented graphically and discussed. Known particular cases are described using passages to the limits.